Tether supports for tethering dogs to bicycles (bikes) for exercise are known. Most seem to employ a rigid bar or frame extending to the side or rear of the bike, with an attachment for a leash. Some have flexible joints or spring members, apparently to absorb shock.
In my experience and opinion, prior tether supports for bikes are often uncomfortable and difficult to use for both dog and rider, interfering with their respective running and riding. A tether support with a long, unsupported leash may be suitable for easy rides on paved roads and sidewalks, but is generally unsuitable for trail riding and mountain biking. A short, rigid tether support might help keep a dog on-trail along narrow paths, but can jerk both dog and rider while moving when their speed and direction do not match, and can jolt the dog during sudden stops.